Heat and mass transfer in nanofluid thin film over an unsteady stretching sheet using Buongiorno s model

Transcription

1 Eur. Phys. J. Plus (2016) 131: 16 DOI /epjp/i Regular Article THE EUROPEAN PHYSICAL JOURNAL PLUS Heat and mass transfer in nanofluid thin film over an unsteady stretching sheet using Buongiorno s model M. Qasim 1,a, Z.H. Khan 2, R.J. Lopez 3, and W.A. Khan 4 1 Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad, Islamabad, Pakistan 2 Department of Mathematics, University of Malakand, Khyber Pakhtunktwo 18800, Pakistan 3 3Maplesoft, Waterloo, Ontario, N2V 1K8, Canada 4 Department of Mechanical and Industrial Engineering College of Engineering, Majmaah University, Saudi Arabia Nomenclature Received: 3 July 2015 / Revised: 11 November 2015 Published online: 22 January 2016 c Società Italiana di Fisica / Springer-Verlag 2016 Abstract. The heat and mass transport of a nanofluid thin film over an unsteady stretching sheet has been investigated. This is the first paper on nanofluid thin film flow caused by unsteady stretching sheet using Buongiorno s model. The model used for the nanofluid film incorporates the effects of Brownian motion and thermophoresis. The self-similar non-linear ordinary differential equations are solved using Maple s built-in BVP solver. The results for pure fluid are found to be in good agreement with the literature. Present analysis shows that free surface temperature and nanoparticle volume fraction increase with both unsteadiness and magnetic parameters. The results reveal that effect of both nanofluid parameters and viscous dissipation is to reduce the heat transfer rate. B 0 Uniform magnetic field u, v Velocity components along the x b A positive constant and y directions, respectively C Nanoparticle volume fraction x, y Cartesian coordinates along the sheet C w Nanoparticle volume fraction at sheet and normal to it, respectively. C fx Skin friction coefficient D B Brownian diffusion coefficient Greek symbols D T Thermophoretic diffusion coefficient α Thermal diffusivity of the base fluid Ec Eckert number β Dimensionless film thickness f(η) Dimensionless stream function φ Dimensionless nanoparticle volume fraction h(t) Film thickness η Similarity variable k Thermal conductivity of base fluid θ Dimensionless temperature M Hartmann number μ Dynamic viscosity of the base fluid Nb Brownian motion parameter ν Kinematic viscosity of the base fluid Nt Thermophoresis parameter σ Electrical conductivity Nu x Local Nusselt number ρ f Density of base fluid Pr Prandtl number ρ p Nanoparticle mass density p Pressure (ρc) f Heat capacity of the base fluid Re x Local Reynolds number (ρc) p Heat capacity of the nanoparticle material S Unsteadiness parameter τ (ρc) p /(ρc) f Sc Schmidt number ψ Stream function Sh x Local Sherwood number T Fluid temperature T w Temperature at the sheet T r Reference temperature Velocity of the stretching sheet U w a mq qau@yahoo.com

2 Page 2 of 11 Eur. Phys. J. Plus (2016) 131: 16 Table 1. Literature on nano-liquid film hydrodynamic models. Authors Models Film thickness Magnetic field characteristics Viscous dissipation Narayana and Sibanda [21] Homogeneous Considered Not considered Not considered Xu et al. [22] Homogeneous Not considered Not considered Not considered Present study Buongiorno s Considered Considered Considered 1 Introduction The heat and mass transport within a liquid film is an attractive area of research because of a wide range of engineering applications such as food processing, insulating materials, polymer, reactor fluidization, heat exchangers, cooling industry and so on. The main application of liquid film flows are in coating industries as wire coating and fiber coating. Any coating process demands an ideal rate of heat and mass transfer within the liquid film to achieve low friction, best appearance, optimum performance and strength. In the pioneer work, Wang [1] investigated the hydrodynamics of a thin liquid film over an unsteady stretching sheet. The unsteady Navier-Stokes equations were transformed to self-similar nonlinear ordinary differential equations. It was shown that a solution exists only in the interval [0, 2] of the unsteady parameter (S). In the case, when S 0 then the fluid has an infinitely thick layer. The other limiting case S 2 represents a liquid film of an infinitesimal thickness. Since then many researchers, including Anderson et al. [2], Chen [3,4], Wang [5], Dandapat et al. [6], Liu and Andersson [7,8], Abel et al. [9], and Nandeppanavar [10], have extended Wang s [1] idea by considering different hydrodynamic and thermodynamic models. The convective heat transfer is poor in ordinary fluids like water, ethylene glycol and mineral oils because of their low thermal conductivity. An increase in the thermal conductivity of the ordinary fluids increases the productivity of the associated heat transfer process. Keeping these facts in mind, many methods are developed for improvement in thermal conductivity of ordinary fluids. One way among these is the suspension of nanoparticles (diameter less than 50 nm) in the ordinary fluids, the term nano defined initially by Choi [11]. Measurements for thermal conductivity have been presented by many researchers with different nanoparticles volume fraction, material and dimension in several ordinary fluids. They all observed that nanofluids have relatively higher thermal conductivity than the ordinary fluids. In these observations, the consideration of Brownian motion (random motion of particles) of the nanoparticles inside the traditional fluids is one mechanism by which an increase in the thermal conductivity of nanofluids is possible. An excellent review of convective transport in nanofluids was presented by Buongiorno [12] and Kakac and Pramuanjarocnkij [13]. Several researchers studied boundary layer flow of nanofluid over stretching sheet using Buongiorno s model. The model used for the nano-liquid film incorporates the effects of Brownian motion and thermophoresis [14 20]. From a hydrodynamic point of view, the concept of nanofluid films is very new and so far only two studies are reported in the literature (see table 1). Narayana et al. [21] studied the effect of nanoparticles on the liquid film considering film thickness. An earlier study by Xu et al. [22] considered a homogeneous model ignoring film thickness. The purpose of this study is to develop a mathematical model for nano-liquid film subject to Brownian motion and thermophoresis. The analysis has been addressed in the presence of viscous dissipation and a transverse magnetic field. The viscous dissipation controls the internal friction in determination of temperature distribution. Industrial applications such as cooling of nuclear reactors, chemical and food processing, oil exploration and bioengineering are based on this significant characteristic. Viscous dissipation changes the temperature distribution by playing a central role like an energy source [23]. Further, electrically conducting fluid has importance in power generators, MHD accelerators, the design of heat exchanges and electrostatic filters and in the field of aerodynamics to control the boundary layer flow. In stretching flows such consideration is very important especially in metallurgical processes including the cooling of continuous strips and filaments drawn through a quiescent fluid and purification of molten metals from non-metallic inclusions. In these processes the quality of final product is strongly influenced by the rate of cooling. The cooling rate is controlled by drawing strips in an electrically conducting fluid subject to a magnetic field [24]. The governing partial differential equations are first transformed into coupled nonlinear ordinary differential equations which are then solved numerically by Maple s built-in dsolve solver. Effects of physical parameters of interest such as magnetic parameter M, Prandtl number Pr, Eckert number Ec, Schmidt number Sc, Brownian motion parameter Nb, thermophoresis parameter Nt, dimensionless film thickness β and the unsteadiness parameter S are made and analyzed. Numerical values of skin friction coefficient, local Nusselt number and local Sherwood number are given and examined. The numerical results were also compared with those reported by Wang [1] and Narayana and Sibanda [22] and we found an excellent agreement with them.

3 Eur. Phys. J. Plus (2016) 131: 16 Page 3 of 11 Fig. 1. Schematic representation of the proposed nano-liquid film model. 2 Problem formulation We consider the thin film flow of a nanofluid over an unsteady stretching sheet which is being stretched with a linear velocity U w = bx 1 γt, where b and γ are positive constants and x is the coordinate measured from origin O (see fig. 1). The effect of an induced magnetic field is negligible under the assumption that a uniform magnetic field of intensity B 0 and small magnetic Reynolds ( number ) is applied in the positive y direction, normal to the surface. ( The) temperature distribution T w (x, t) =T 0 T r (1 γt) 3/2 and nanoparticle volume fraction C w (x, t) =C 0 C r (1 γt) 3/2 bx 2 2ν on the surface are assumed to vary with the distance x from the slit; T 0 and C 0, respectively, are the temperature and nanoparticle volume fraction of nanofluid at the slit, T r and C r are the constant reference temperature and the constant reference nanoparticle volume fraction respectively, such that (0 <T r <T 0, 0 <C r <C 0 ). It is further assumed that the base fluid and the suspended nanoparticles are in thermal equilibrium. The transverse magnetic field B(t) is defined by [9] B(t) =B 0 (1 γt) 1/2. (1) Under the above assumptions, the governing boundary layer equations of the conservation of mass, momentum, energy and nanoparticles fraction are [12 20] u x + v y =0, (2) u t + u u x + v u y = ν 2 u y 2 σb2 (t) u, (3) ρ f T t + u T x + v T y = T α 2 y 2 + ν ( ) { 2 ( ) ( )( ) } 2 u C T DT T + τ D B +, (4) c p y y y T y C t + u C x + v C ( 2 ) ( )( y = D C DT 2 ) T B y 2 + y 2, (5) where u and v are the velocity components along the x and y directions, respectively, α = T k (ρc) f bx 2 2ν is the thermal diffusivity of the base fluid, τ = (ρc)p (ρc) f is the ratio of nanoparticle heat capacity and the base fluid heat capacity, C is the local nanoparticle volume fraction, D B is the Brownian diffusion coefficient, D T is the thermophoretic diffusion coefficient, T is the local temperature, ρ is the density and σ is the electrical conductivity. The associated boundary conditions are [1 10] u = U w, v =0, T = T w, C = C w at y =0, (6) u y = T y = C y =0, v = h t at y = h(t), (7)

4 Page 4 of 11 Eur. Phys. J. Plus (2016) 131: 16 where h(t) is the film thickness. We define the following similarity transformations [21]: ( ) 1/2 b η = y, ν(1 γt) ( ) 1/2 νb ψ(x, y, t) = xf(η), 1 γt( ) bx 2 T (x, y, t) =T 0 T r (1 γt) 3/2 θ(η), 2ν ( ) bx 2 C(x, y, t) =C 0 C r (1 γt) 3/2 φ(η), 2ν (8) where the stream function ψ(x, y) is defined as u = ψ y, v = ψ x. The dimensionless film thickness β is defined as [21, 22] ( ) 1/2 b β = h(t). (9) ν(1 γt) Equation (9) gives dh dt = γβ 2 ( ν b ) 1/2 (1 γt) 1/2. (10) On substituting eq. (8) into eqs. (2) to (5), we obtain the following coupled system of ordinary differential equations: ( f + ff f 2 S f + η 2 f ) Mf =0, (11) 1 Pr θ + fθ 2f θ S 2 (3θ + ηθ )+Ecf 2 + Ntθ 2 + Nbφ θ =0, (12) ( φ + Sc fφ 2f φ S ) 2 (3φ + ηφ ) + Nt Nb θ =0. (13) where prime denotes the differentiation with respect to similarity variable η and dimensionless parameters S, M, Pr, Ec, Sc, Nb and Nt are defined as S = γ b, M = σb2 0 ρb, Pr = ν α, Ec = Uw 2 c p (T w T 0 ), Sc = ν, D B Nb = τd B(C w C ), Nt = τd } T (T w T ). ν νt (14) The transformed boundary conditions are as follows: f(0) = 0, f (0) = 1, f(β) = Sβ 2, f (β) =0, θ(0) = φ(0) = 1, θ (β) =φ (β) =0. (15) The physical quantities of interest are the skin friction coefficient C f, the local Nusselt number Nu x and the local Sherwood number Sh x, which are defined as: C fx = τ w q w x 1 2 ρu, Nu = w 2 k(t w T 0 ), Sh = q m x D B (C w C 0 ) where τ w = μ( u/ y) y=0, q w = k( T/ y) y=0,andq m = D B ( C/ y) y=0, are the shear stress, heat and mass fluxes at the surface, respectively. Using the variables in (8), the associated expressions for dimensionless skin-friction coefficient C f, reduced Nusselt number θ (0) and reduced Sherwood number φ (0) are defined as Re 1/2 x C fx = f (0), Re 1/2 x Nu x = θ (0), Re 1/2 x Sh x = φ (0), (17) where Re x = U w x/ν is the local Reynolds number based on the stretching velocity. (16) 3 Method of solution All of the numeric values and graphs obtained in this study are obtained by solving the nonlinear system of differential equations (11)-(13) numerically subject to the initial and boundary conditions in (15). First the boundary value

5 Eur. Phys. J. Plus (2016) 131: 16 Page 5 of 11 Table 2. Comparison of values of film thickness β and skin friction coefficient f (0) for various value of S. Note: Wang [1] has used a different similarity transformation in his paper, so that f (0) in his paper is the same as βf (0) in the present study. S Wang [1] Narayana and Sibanda [21] Present results β f (0) β βf (0) β βf (0) problem is converted to initial value problem and writes the differential equations as a first-order system: df 0 dη = f 1, df 1 dη = f 2, df ( 2 dη = S f 1 + η ) 2 f 2 + Mf 1 + f1 2 f 0 f 2, dθ 0 dη = θ 1, dφ 0 dη = φ 1, dθ 1 ( f 0 θ 1 +2f 1 θ 0 + S2 ) (3θ 0 + ηθ 1 ) Ecf 22 Ntθ 21 Nbφ 1 θ 1 dη =Pr dφ 1 dη = Sc ( f 0 φ 1 2f 1 φ 0 S ) 2 (3φ 0 + ηφ 1 ) Nt dθ 1 Nb dη., (18) The initial and boundary conditions (15) after decomposition take the following form: f 0 (0) = 0, f 1 (0) = 1, θ 0 (0) = φ 0 (0) = 1, (19) f 2 (β) =0, θ 1 (β) =φ 1 (β) =0, (20) f 0 (β) = Sβ 2. (21) To solve resulting IVP (18) subject to the conditions (19) and (20), the values for f 2 (0), θ 1 (0) and φ 1 (0) are needed. However, no such numerical values are known prior. Therefore, the initial guesses of f 2 (0), θ 1 (0) and φ 1 (0) are chosen and the maple built-in dsolve package is used to obtain solution for a set of governing parameters and a known value of S. On trial and error basis, the value of the film thickness β is adjusted such that the condition (21) holds. The step-size is taken as Δη =0.01, and the accuracy to the fifth decimal place is regard as the criterion of convergence. 4 Results and discussion The comparison of present results of film thickness, skin friction, surface temperature and Nusselt numbers with the published data is presented in tables 2 and 3. The present results are found to be in an excellent agreement, which shows the accuracy of our results. It is noticed that the film thickness decreases and skin friction increases with an increase in unsteadiness parameter (table 2). The behavior of surface temperature and Nusselt number with Prandtl numbers for different values of unsteadiness parameter is reported in table 3. It reveals that due to an increase in Nusselt numbers, the surface temperature decreases with Prandtl numbers. This is because the thermal boundary layer thickness decreases with an increase in Prandtl numbers. The effects of magnetic field on the dimensionless axial

6 Page 6 of 11 Eur. Phys. J. Plus (2016) 131: 16 Table 3. Comparison of values of surface temperature and wall temperature gradient for pure fluid. Note: Wang [1] has used a different similarity transformation in his paper, so that θ (0) in his paper is the same as βθ (0) in the present study. Pr Wang [1] Narayana and Sibanda [21] Present results θ(1) θ (0) θ(β) βθ (0) θ(β) βθ (0) S = S = Fig. 2. Effect of magnetic M and unsteady parameters S on the axial velocity, temperature and nanoparticle volume fraction profiles with Ec =0.1, Nt =0.5, Nb =0.5, Sc =1,Pr=3.97. velocity, temperature and nanoparticle volume fraction are displayed in figs. 2(a)-(c), respectively, for two different values of unsteadiness parameter. In the absence of a magnetic field, the dimensionless axial velocity is found to be higher in the hydrodynamic boundary layer and it decreases with an increase in the magnetic field, as shown in fig. 2(a). In fact, the magnetic field produces a Lorentz force that opposes the motion and hence decreases the velocity. The numeric solution stops when it satisfies the boundary condition at the free surface corresponding to a certain film thickness. This film thickness also decreases for each value of the unsteadiness parameter. As a result, the boundary layer thickness decreases with an increase in the magnetic field and unsteadiness. The dimensionless temperature is higher at the surface and it decreases with the transverse distance inside the thermal boundary layer, as shown in fig. 2(b). Due to a decrease in the dimensionless film thickness, the dimensionless temperature increases with the

7 Eur. Phys. J. Plus (2016) 131: 16 Page 7 of 11 Fig. 3. Effect of the thermophoresis Nt and Brownian motion Nb parameters on the temperature and nanoparticle volume fraction profiles. Fig. 4. Effect of the Eckert, Prandtl and Schmidt numbers on the temperature and nanoparticle volume fraction profiles. magnetic field for each value of unsteadiness parameter and as a result, the thermal boundary layer thickness increases. The nanofluid film thickness also decreases with both parameters. The effects of magnetic field and unsteadiness parameter on the dimensionless nanoparticle volume fraction are reported in fig. 2(c). The effects of nanofluid parameters on the dimensionless temperature and nanoparticle volume fraction in the presence of the magnetic field are displayed in figs. 3(a) and 3(b), respectively. Acetone (Pr = 3.97) is selected in this case. It is deliberated that the nanofluid parameters have almost negligible effects on the dimensionless temperature (fig. 3a). The film thickness is based on the thermal boundary condition at the free surface. Figure 3(b) depicts the effects of the same parameters on the dimensionless nanoparticle volume fraction. It is noticed that the dimensionless nanoparticle volume fraction decreases with the Brownian motion parameter and increases with the thermophoresis parameter. The effects of dimensionless numbers on the dimensionless temperature and nanoparticle volume fraction are reported in figs. 4(a) and 4(b), respectively, for four different fluids. An increase in Prandtl number means a decrease of fluid thermal conductivity which causes the reduction of the thermal boundary layer thickness. The film thickness remains fixed for a specific value of unsteadiness parameter, as shown in fig. 4(a). In the absence of viscous dissipation, the thermal boundary layer thickness is minimal and increases with viscous dissipation. Figure 4(b) illustrates the effects of the Schmidt number on the dimensionless nanoparticle volume fraction. It is shown that the boundary

8 Page 8 of 11 Eur. Phys. J. Plus (2016) 131: 16 Fig. 5. Variations in transverse free surface velocity, free surface temperature and nanoparticle volume fraction with the magnetic and unsteady parameters. Fig. 6. Variations in transverse free surface temperature and free surface nanoparticle volume fraction with the Eckert number, thermophoresis and Brownian motion parameters. layer thickness decreases with an increase in the Sherwood number, which identifies fluid flows where simultaneous momentum and mass diffusion convection processes take place. It physically measures the relative thickness of the hydrodynamic layer and mass transfer boundary layer. It is important to note that, for larger values of Schmidt numbers, the viscous dissipation effects on the dimensionless nanoparticle volume fraction are negligible. The variation in the free surface velocity, temperature and nanoparticle volume fraction with the magnetic and unsteadiness parameters is shown in figs. 5(a)-(c) respectively. Figure 5(a) reveals that the free surface velocity decreases with increasing magnetic field. This is due to the Lorentz force generated by the magnetic field. Moreover, the free surface velocity is found to be maximal when S = 0 and it decreases as S increases. Figure 5(b) shows the variation of the free surface temperature with the transverse distance. It is noticed that the temperature diminishes smoothly with S for any value of the magnetic field. This is because the free surface temperature is higher in nanoparticle volume fraction than that in the thermal boundary layer. Figure 5(c) illustrates the same behavior of the free surface nanoparticle volume fraction with magnetic and unsteadiness parameters. The effects of nanofluid parameters and Eckert number on the free surface temperature and nanoparticle volume fraction are depicted in figs. 6(a) and (b). Both nanofluid parameters and Eckert number help in enhancing the free surface temperature, as shown in fig. 6(a). In fact, the Eckert number characterizes the dissipation. The thermal boundary layer thickness increases by increasing the Eckert number Ec. This is because of the fact that due to large viscous resistance there is more accumulation of heat energy in the fluid particles near the boundary. In the absence of viscous dissipation, the free surface temperature is found to be smaller and increases with an increase in viscous dissipation. The free surface

9 Eur. Phys. J. Plus (2016) 131: 16 Page 9 of 11 Fig. 7. Variations in skin friction coefficient, Nusselt and Sherwood numbers with the magnetic and unsteady parameters. Table 4. Values of surface temperature and Nusselt number for various values of the governing parameters with Pr = 6.2 and Sc = 10. M Ec Nb θ(β) θ (0) S =0.8 S =1.2 S =0.8 S =1.2 Nt =0.1 Nt =0.3 Nt =0.1 Nt =0.3 Nt =0.1 Nt =0.3 Nt =0.1 Nt = also depends upon the nanofluid parameters and Eckert number. The Brownian motion parameter tends to decrease the free surface nanoparticle volume fraction but the thermophoresis parameter helps in increasing the free surface nanoparticle volume fraction. Like the free surface temperature, the nanoparticle volume fraction also increases with viscous dissipation. Variations with the magnetic and unsteadiness parameters for skin friction, Nusselt and Sherwood numbers are reported in figs. 7(a)-(c), table 4 and table 5, respectively. Since the magnetic field produces a Lorentz force that opposes the motion and decreases the velocity inside the boundary layer, the skin friction increases with an increase in the magnetic field. For smaller values of the unsteadiness parameter, this increase can be seen clearly but, for larger values of unsteadiness parameter, the skin friction decreases and rapidly approaches zero. The other disadvantage of a magnetic field is the decrease in Nusselt numbers with an increase in the magnetic field, as shown in fig. 7(b). It is also noticed that the Nusselt numbers increase with the unsteadiness parameter and then drops quickly. The same behavior could be observed for the Sherwood numbers in fig. 7(c). The effects of nanofluid parameters on the Nusselt and Sherwood numbers are presented in figs. 8(a) and (b) for two different values of Eckert number. As expected, the nanofluid parameters tend to reduce Nusselt numbers for both values of the Eckert number. In the absence of dissipation, the Nusselt numbers are found to be higher and they decrease with an increase in dissipation. The Sherwood numbers decrease with increasing thermophoresis effect but Brownian motion helps in increasing the Sherwood numbers. In the absence of dissipation, the decrease in Sherwood numbers is more pronounced than in the presence of viscous dissipation.

10 Page 10 of 11 Eur. Phys. J. Plus (2016) 131: 16 Table 5. Values of surface nano-nanoparticle volume fraction and Sherwood number for various values of the governing parameters with Pr = 6.2 andsc = 10. M Ec Nb φ(β) φ (0) S =0.8 S =1.2 S =0.8 S =1.2 Nt =0.1 Nt =0.3 Nt =0.1 Nt =0.3 Nt =0.1 Nt =0.3 Nt =0.1 Nt = Fig. 8. Effects of Brownian motion and thermophoresis parameters on Nusselt and Sherwood numbers. 5 Conclusions In this study, heat and mass transfer in a nanofluid film over an unsteady stretching sheet are investigated using Buongiorno s model. A similarity solution is presented to explore the effects of film thickness, unsteadiness, magnetic field, viscous dissipation and nanofluid parameters on the fluid flow and heat and mass transfer rates. The key findings are: Free surface velocity decrease with both unsteadiness and magnetic parameters. The dimensionless film thickness increases with both unsteadiness and magnetic parameters. Viscous dissipation enhances the thermal boundary layer thickness. Free surface temperature and nanoparticle volume fraction increase with both unsteadiness and magnetic parameters. For smaller values of unsteadiness, the skin friction as well as dimensionless heat and mass transfer rates increase. The dimensionless heat transfer rates decrease with both nanofluid parameters and viscous dissipation.

11 Eur. Phys. J. Plus (2016) 131: 16 Page 11 of 11 References 1. C.Y. Wang, Quart. Appl. Math. 48, 601 (1990). 2. H.I. Andersson, J.B. Aarseth, B.S. Dandapat, Int. J. Heat Mass Transf. 43, 69 (2000). 3. C.H. Chen, Heat Mass Transf. 39, 791 (2003). 4. C.H. Chen, J. Non-Newtonian Fluid Mech. 135, 128 (2006). 5. C. Wang, Heat Mass Transf. 42, 759 (2006). 6. B.S. Dandapat, B. Santra, H.I. Andersson, Int. J. Heat Mass Transf. 46, 3009 (2003). 7. I.C. Liu, H.I. Andersson, Int. J. Heat Mass Transf. 51, 4018 (2008). 8. I.C. Liu, H.I. Andersson, Int. J. Thermal Sci. 47, 766 (2008). 9. M.S. Abel, N. Mahesha, J. Tawade, Appl. Math. Model. 33, 3430 (2009). 10. M.M. Nandeppanavar, K. Vajravelu, M.S. Abel, S. Ravi, H. Jyoti, Int. J. Heat Mass. Transf. 55, 1316 (2012). 11. S.U.S. Choi, Enhancing thermal conductivity of fluids with nanoparticles, inthe Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, Vol. 66 (ASME, San Francisco, USA, 1995), pp J. Buongiorno, ASME J. Heat Transf. 128, 240 (2006). 13. S. Kakaç, A. Pramuanjaroenkij, Int. J. Heat Mass Transf. 52, 3187 (2009). 14. W.A. Khan, I. Pop, Int. J. Heat Mass Transf. 53, 2477 (2010). 15. M.A. Hamad, I. Pop, Transp. Porous Med. 87, 25 (2011). 16. O.D. Makinde, W.A. Khan, Z.H. Khan, Int. J. Heat Mass Transf. 62, 526 (2013). 17. M.A.A. Hamad, M. Ferdows, Commun. Nonlinear Sci. Numer. Simulat. 17, 132 (2012). 18. P. Rana, R. Bhargava, Commun. Nonlinear. Sci. Numer. Simulat. 17, 212 (2012). 19. R. Kandasamy, P. Loganathan, P.P. Arasu, Nucl. Eng. Des. 241, 2053 (2011). 20. A.B. Rosmila, R. Kandasamy, I. Muhaimin, Appl. Math. Mech. 33, 593 (2012). 21. M. Narayana, P. Sibanda, Int. J. Heat Mass. Transf. 55, 7552 (2012). 22. H. Xu, I. Pop, X.C. You, Int. J. Heat Mass. Transf. 60, 646 (2013). 23. R. Cortell, Phy. Lett. A 372, 631 (2008). 24. C.H. Chen, Int. J. Non-Linear Mech. 44, 596 (2009).